The proof strategy is as follows. Since \(\left( \Phi _{ef}^{\ast }(p),k+1\right) >1\) , and since \(k+1\)  divides the length of each \(G_{x}\) -orbit distinct from \(\left\lbrace  x\right\rbrace  \)  by Lemma REF (3), we derive that \(\Phi _{ef}^{\ast }(p)\mid \left|X_{x}\right|\)  and, moreover, that \(X_{x}\)  contains a Singer cyclic subgroup of \(X\)  for \(e=n\)  by using simple
group-theoretical arguments. Then \(X_{x}^{^}\)  is a subgroup
of \(SL_{n}(q)\)  classified in Theorem 3.1
of [1]} for \(e<n\) . For \(e=n\) , we use instead the classification for \(X_{x}^{^}\)  contained in [2]}. By combining the fact that the candidates \(X_{x}\)  must fulfill \(\left[ X:X_{x}\right]=k^{2}\)  with the information contained in Lemma 3.13 of [3]}, we show that \(\left|\mathrm {Out(}X\mathrm {)}\right|\geqslant \left|X_{x}\right|_{p}\) . Hence, \(X_{x}\)  is a large subgroup of \(X\)  by Lemma REF (2).
